Divisor then it is a Direct Rule As In the following Questions Quest. 1. If 8 Labourers can do a certain piece of work in 12 dayes in how many dayes will 16 Labourers do the same Answer in 6 dayes Having placed the numbers according to the 6 Rule of the 10th Chapter I consider that if 8 men can finish the work in 12 dayes 16 men will do it in lesser or fewer dayes then 12 therefore the biggest Extream must be the Divisor which is 16 and therefore it is the Rule of 3 Inverse wherefore I multiply the first and second numbers together viz. 8 by 12 and their product is 96 which divided by 16 Quotes 6 days for the Answer and in so many days will 16 Labourers perform a piece of work when 8 can do it in 12 days Quest. 2. If when the measure viz. a peck of wheat cost 2 shillings the peny Loaf weighed according to the Standard Statute or Law of England 8 ounces I demand how much it will weigh when the peck is worth 1 s. 6 d. according to the same Rate or Proportion Answer 10 oz. 13 p.w. 8 grs. Having placed and reduced the given Numbers according to the 6 and 9 rules of the 10th Chapter I consider that at 1 s. 6 d. per Peck the peny Loaf will weigh more then at 2 s. per Peck for as the price decreaseth the weight Increaseth and as the price Increaseth so the weight diminisheth wherefore because the term Requireth more then the second the lesser Extream must be the Divisor viz. 1 s. 6 d. or 18 peâ—â—e and having finished the work I find the Answer to be 10 oz. 13 p.w. 8 gr and so much will the peny Loaf weigh when the peck of wheat is worth 1 s. 6 d. according to the given Rate of 8 ounces when the peck is worth 2 shillings the work is plain in the following operation Quest. 3. How many pieces of money or Merchandize at 20 s. per piece are to be given or Received for 240 pieces the value or price of every piece being 12 shillings Answer 144. For if 12 s. Require 240 pieces then 20 shillings will Require less therefore the biggest Extream must be the Divisor which is the 3 number c. See the work Quest. 4. How many yards of 3 quarters broad are Required to double or be equal in measure to 30 yards that are 5 quarters broad Answer 50 yards For say If 5 quarter wide Require 30 yards long what length will three quarters broad require here I consider that 3 quarters broad will Require more yards then 30 for the narrower the cloth is the more in length will go to make equal measure with a broader piece Quest. 5. At the Request of a friend I lent him 200 l. for 12 moneths promising to do me the like Curtesie at my necessity but when I came to Request it of him he could let me have but 150 l. now I desire to know how long I may keep this money to make plenary Satisfaction for my former kindness to my Friend Answer 16 Moneths I say If 200 l. Require 12 Moneths what will 150 l. Require 150 l. will Require more time then 12 Moneths therefore the lesser extream viz. 150 must be the Divisor Multiply and Divide and you will find the 4th inverted Proportional to be 16 and so many Moneths I ought to keep the 150 l. for satisfaction Quest. 6. If for 24 s. I have 1200 l. weight carried 36 M. how many M. shall 800 l. be carried for the same mony Answer 24 M. Quest. 7. If for 24 s. I have 1200 l. carried 36 Miles how many pound weight shall I have carried 24 miles for the same money Answer 800 l. weight Quest 8. If 100 workmen in 12 dayes finish a piece of work or service how many workmen are sufficient to do the same in 3 dayes Answer 400 workmen Quest. 9. A Colonel is besieged in a Town in which are 1000 Souldiers with provision of Victuals only for 3 Moneths the Question is how many of his Souldâ—ers must he dismâ—ss that his Victuals may last the Remaining Souldiers 6 Moneths Answer 500 he must keep and dismiss as many Quest. 10. If wine worth 20 l. is sufficient for the ordinary of 100 men when the Tun is sold for 30 l. how many men will the same 20 pounds worth suffice when the Tun is worth 24 l. Answer 125 men Quest. 11. How much plush is sufficient to line a Cloak which hath in it 4 yards of 7 quarters wide when the Plush is but 3 quarters wide Answer 9 â…“ yds of Plush Quest. 12. How many yards of Canvas that is Ell wide will be sufficient to line 20 yards of Say that is 3 quarters wide Answer 12 yards Quest. 13. How many yards of matting that is 2 foot wide will cover a floor that is 24 foot long and 20 foot broad Answer 240 foot Quest. 14. A Regiment of Souldiers consisting of 1000 are to have new Coats and each coat to contain 2 yds 2 qrs of Cloth that is 5 qrs wide and they are to be lined with Shalloon that is 3 quarters wide I demand how many yards of Shalloon will line them Answer 16666 â…” yards Quest. 15. A Messenger makes a Journey in 24 dayes when the day is 12 hours long I desire to know in how many dayes he will go the same when the day is 16 hours long Answer in 18 dayes Quest. 16. Borrowed of my friend 64 l. for 8 Moneths And he hath occasion another time for to borrow of me for 12 Months I desire to know how much I must lend to make good his former kindness to me Answer 42 l. 13 s. 4 d. 4. The General Effect of the Rule of 3 Inverse is contained in the Definition of the same that is to find a fourth term in a Reciprocal Proportion inverted to the Proportion given The second Effect is by two prises or values of two several pieces of money or Merchandize known to find how many pieces of the one price is to be given for so many of the other And consequently to Reduce or Exchange one sort of Money or Merchandize into another Or contrariwise to find the price unknown of any piece given to Exchange in Reciprocal Proportion The third Effect is by two dâ—ffering prizes of a measure of wheat bought or sold and the weight of the Loaf of bread made answerable to one of the prises of the measure given to find out the weight of the same Loaf answerable to the other price of the said measure given Or contrariwise by the two several weights of the same prized Loaf and the price of the measure of wheat answerable to one of those weights given to find out the other price of the measure answerable to the other weight of the same Loaf The Fourth Effect is by two lengths and one breadth of two Rectangular planes known to find out another breadth unknown Or by two
are Commensurable because three measures them both CHAP. II. Of the Natural Division of Integers and the several Denominations of their parts 1. BEfore we come to Calculation or the ordering of Numbers to operate any Arithmetical Question proposed we will lay down Tables of the Denomination of several Integers and after that having mentioned the several Species or kinds of Arithmetick we shall immediately handle the Species of Numeration which are the main Pillars upon which the whole Fabrick of this Art is built Of Money Weights c. 2. The least Denomination or Fraction of money used in England is a farthing from whence is produced the following Tables called the Tables of Coyn viz.    and therefore 1 farthing make 1 farthing l. s. d. qrs 4 farthings 1 Penny 1 20 12 4 12 Pence 1 shilling 1 20 240 960 20 Shillings 1 Pound  1 12 48      1 4 The first of these Tables viz. that on the left hand is plain and easie to be understood and therefore wants no directions In the second Table above the line you have 1 l. 20 s. 12 d. 4 qts whereby is meant that 1 pound is equal to 20 shillings and one shilling is equal to 12 pence and one peny equal to 4 farthings under the line is 1 l. 20 s. 240 d. 960 qts which signifies one pound to contain 20 shillings or 240 pence or 960 farthings in the second line below that is 1 s. 12 d. 48 qts the first standing under the denomination of shillings whereby is to be noted that one shilling is equal to 12 pence or 48 farthings and likewise that below that one peny is equal in value to four farthings understand the like reason in all the following Tables of weight measure time motion and dozen Troy weight 3. The least Fraction or Denomination of weight used in England is a grain of wheat gathered out of â—â—e middle of the ear and well dryed from whence are produced these following â—â—bles of weight called Troy weight 32 Grains of wheat make 24 artificial grains 24 Artificial graines 1 Penny weight 20 Peny-weight 1 Ounce 12 Ounces 1 Pound Troy weight And Therefore l. oun dra grains 1 12 20 24 1 12 240 5760  1 20 480   1 24 Troy Weight serveth only to weigh Bread Gold silver and Electuaries it also regulateth and prescribeth a form how to keep the money of England at a certain standard The Goldsmiths have devided the ounce Troy weight into other parts which they generally call mark weight the denominative parts thereof are as followeth viz. A mark being an ounce Troy is divided into 24 equal parts called Carects and each Carect into 4 grains so that in a mark are 96 Grains by this weight they distinguish the different finess of their Gold for if to the finest of Gold be put 2 Carects of Alloy which is of Silver Copper or other baser mettle with which they use to mix their Gold or silver to abate the finess thereof both making when cold but an ounce or 24 Carects then this Gold is said to be 22 Carects fine for if it come to be Refined the 2 Carects of alloy will fly away and leave only 22 Carects of pure Gold the like to be considered of a greater or lesser quantity And as the finess of gold is estimated by Carects so the finess of silver is distinguished by ounces for if a pound of it be pure and looseth nothing in the Refining such silver is said to be twelve ounces fine but if it looseth any thing it is said to contain so much fineness as the loss wanteth of 12 ounces as if it lose an ounce it is said to be 11 ounces fine and if it lose one ounce 14 peny weight then it is said to be 10 ounces 6 peny weight fine and that which loseth two ounces four penny weight 16 grains is said to be nine ounces 15 penny weight 8 grains fine c. the like of a greater or lesser quantity Apothecaries weights 4. The Apothecaries have their weights deduced from Troy weight a pound Troy being he greatest Integer a Table of whose division and sub-division followeth viz. 1 pound makes 12 ounces And therefore 1 ounce 8 drams l. oun dram scrup gr 1 dram 3 scruples 1 12 8 3 20 1 scruple 20 grains 1 12 96 218 5760     1 8 24 480      1 3 60       1 20 5. Thus much concerning Troy weight and its derivative weights which as was said before serveth to weigh Bread Gold Silver and Electuaries now besides Troy weight there is another kind of weight used in England commonly known by the name of Averdupois weight a pound of which is equal to 14 ounces 12 peny weight Troy weight and it serveth to weigh all kindes of Grocery wares as also butter Cheese Flesh Wax Tallow Rozen Pitch Lead and all such kind of garbel the Table of which weight is as followeth The Table of Averdupois weight 4 quarters of a dram makes one dram 16 drams one ounce 16 ounces one pound 28 Pounds 1 quarter of a hundred 4 quarters 1 hund weight or 112 20 Hundred 1 Tun. And therefore Tun C. qts l. oun doz qts 1 20 4 28 16 16 4 1 20 80 2240 13440 215040 860160  1 4 112 1792 28672 114688   1 28 448 7168 28672    1 16 256 1024     1 16 64      1 4 Wooll is weighed with this weight but only the divisions are not the same A Table whereof followeth A Table of the denominative parts of Wool-weight 7 Pounds make 1 Clove 2 Cloves 1 Stone 2 Stone 1 Todd 6 Todd 1 Stone 1 Wey 2 Weyes 1 Sack 12 Sacks 1 Last And Therefore Last Sacks Wey Todd Stone Cloves l. 1 12 2 6 ½ 2 2 7 1 12 24 156 312 624 4368  1 2 13 26 52 364   1 6 ½ 13 26 182    1 2 4 28     1 2 14      1 7 Note that in some Countreys the Wey is 256 l. Averdupois as is the Suffolk Wey But in Essex there is 336 l. in a Wey 6. The least Denominative part of Liquid measure is a Pint and it is taken from Troy weight because 1 pound of wheat Troy weight makes 1 Pint liquid A Table of which Measure followeth The Table of Liquid Measure 1 pound of wheat Troy make 1 pint 2 pints 1 quart 2 quarts 1 pottle 2 pottles 1 gallon 8 gallons 1 firk of ale sope or herr 9 gallons 1 firkin of beer 10 gallons and a half 1 firk of Salmon or Eeles 2 firkins 1 Kilderkin 2 Kilderkins 1 barrel 42 gallons 1 Tierce of wine 63 gallons 1 hogshead 2 hogsheads 1 pipe or butt 2 pipes or butts 1 Tunn of wine And Therefore Tun pipes hhds
of three also by 〈◊〉 things of 〈…〉 are reduced to another 〈…〉 any Number of Integers by the price of the Integer the Product will discover the price of the Quantity or Number of Intâ—gers given In a Râ—ctangular Solid if you multiply the bredâ—h of the base by the depth and that Product by the length this last Product will discover the Solidity or content of the same Solid Some Questions proper to this Rule may be these following Quest. 1. What is the content of a square piece of ground whose length is 28 perches and breadth 13 perches Answer 364 square perches for multiplying 28 the length by 13 the breadth the Product is so much Quest. 2. There is a square battail whose Flank is 47 men and the files 19 deep what Number of men doth that battail contain Facit 893 for multiplying 47 by 19 the Product is 893. Quest. 3. If any one thing cost 4 shillings what shall 9 such things cost Answer 36 shillings for multiplying 4 by 9 the Product is 36. Quest. 4. If a piece of Money or Merchandize be worth or cost 7 shillings what shall 19 such pieces of Money or Merchandize cost Facit 133 shillings which is equal to 6 l. 13 s. Quest. 5. If a Souldier or Servant get or spend 14 s. per moneth what is the Wages or Charges of 49 Souldiers or Servants for the same time multiply 49 by 14 the Product is 686 s. for the Answer Quest. 6. If in a day there are 24 hours how many hours are there in a year accounting 365 dayes to constitute the year Facit 8760 hours to which if you add the 6 hours over and above 365 dayes as there is in a year then it will be 8766 hours now if you multiply this 8766 by 60 the Number of Minutes in an hour it will produce 525960 for the Number of Minuts in a Year CHAP. VII Of Division of whole Numbers 1. DIVISION is the Separation or Parting of any Number or Quantity given into any parts assigned Or to find how often one Number is Contained in another Or from any two Numbers given to find a third that shall consist of so many Units as the one of those two given Numbers is Comprehended or contained in the other 2. Division hath three Parts or Numbers Remarkable viz. First the Dividend Secondly the Divisor and Thirdly the Quotient The Dividend is the Number given to be Parted or Divided The Divisor is the Number given by which the Dividend is divided Or it is the Number which sheweth how many parts the Dividend is to be divided into And the Quotient is the Number Produced by the Division of the two given Numbers the one by the other So 12 being given to be divided by 3 or into three equal parts the Quotient will be 4 for 3 is conâ—ained in 12 four times where 12 is the Dividend and 3 is the Divisor and 4 is the Quotient 3. In Division set down your Dividend and draw a Crooked line at each end of it and before the line at the left hand place the Divisor and behind that on the right hand place the figures of the Quotient as in the margent where it is required to divide 12 by 3 First I set down 12 the Dividend and on each side of it do I draw a crooked line and before that on the left hand do I place 3 the Divisor then do I seek how often 3 is contained in 12 and because I find it 4 times I put 4 behind the Crooked line on the Right hand of the Dividend denoting the Quotient 4. But if the Divisor being a single Figure the Dividend consisteth of two or more places then having placed them for the work as is before directed put a point under the first Figure on the left hand of the Dividend provided it be bigger then or equal to the Divisor but if it be lesser then the Divisor then put a point under the second Figure from the left hand of the Dividend which Figures as far as the point goeth from the left hand are to be Reckoned by themselves as if they had no dependance upon the other part of the Dividend and for distinction sake may be called the Dividual then ask how often the Divisor is contained in the Dividual placing the answer in the Quotient then multiply the Divisor by the Figure that you placed in the Quotient and set the product thereof under the Dividual then draw a line under that product and Subtract the said Product from the Dividual placing the Remainder under the said line then put a point under the next figure in the Dividend on the Right hand of that which you put the point before and draw it down placing it on the Right hand of the Remainder which you found by Subtraction which Remainder with the said Figure annexed before it shall be a new dividual then seek again how often the divisor is contained in this new dividual and put the Answer in the Quotient on the Right hand of the Figure there before then multiply the divisor by the last Figure that you put in the Quotient and subscribe the Product under the dividual and make Subtraction and to the Remainder draw down the next Figure from the grand dividend having first put a point under it and put it on the right hand of the Remainder for a new dividual as before c. Observing this general Rule in all kind of Division first to seek how often the divisor is contained in the dividual then having put the answer in the quotient multiply the Divisor thereby and Subtract the Product from the dividual An Example or two will make the Rule plain Let it be Required to divide 2184 by 6 I dispose of the Numbers given as is before directed and as you see in the margent in order to the work then because 6 the divisor is more then 2 the first Figure of the dividend I put a point under 1 the second Figure which make the 21 for the Dividual then do I ask how often 6 the divisor is contained in 21 and because I cannot have it more then 3 times I put 3 in the Quotient and thereby do I multiply the divisor 6 and the product is 18 which I set in order under the dividual and Subtract it therefrom and the Remainder 3 I place in order under the line as you see in the Margent Then do I make a point under the next Figure of the dividend being 8 and draw it down placing it before the Remainder 3 So have I 38 for a new dividual then do I seek how often 6 is contained in 38 and because I cannot have more than 6 times I put 6 in the quotient and thereby do I multiply the divisor 6 and the product 36 I put under the dividual 38 and Subtract it therefrom and the remainder 2 I put under the line as you see in the Margent Then do I put a point under the